Abstract
PHCpack implements numerical algorithms for solving polynomial systems using homotopy continuation methods. In this paper we describe two types of interfaces to PHCpack. The first interface PHCmaple originally follows OpenXM, in the sense that the program (in our case Maple) that uses PHCpack needs only the executable version phc built by the package PHCpack. Following the recent development of PHCpack, PHCmaple has been extended with functions that deal with singular polynomial systems, in particular, the deflation procedures that guarantee the ability to refine approximations to an isolated solution even if it is multiple. The second interface to PHCpack was developed in conjunction with MPI (Message Passing Interface), needed to run the path trackers on parallel machines. This interface gives access to the functionality of PHCpack as a conventional software library.
This material is based upon work supported by the National Science Foundation under Grant No. 0134611 and Grant No. 0410036. Date: 22 June 2006.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Allgower, E.L., Georg, K.: Introduction to Numerical Continuation Methods. Classics in Applied Mathematics, vol. 45. SIAM, Philadelphia (2003)
Anderson, E., Bai, Z., Bischof, C., Blackford, L.S., Demmel, J., Dongarra, J., Du Croz, J., Greenbaum, A., Hammerling, S., McKenny, A., Sorensen, D.: LAPACK User’s Guide, 3rd edn. SIAM, Philadelphia (1999), available online via: http://www.netlig.org/lapack
Corless, R.M., Kaltofen, E., Watt, S.M.: Hybrid methods. In: Grabmeier, J., Kaltofen, E., Weispfenning, V. (eds.) Computer Algebra Handbook, pp. 112–125. Springer, Heidelberg (2002)
Leykin, A., Verschelde, J.: Decomposing solution sets of polynomial systems: a new parallel monodromy breakup algorithm. The International Journal of Computational Science and Engineering (accepted for publication)
Leykin, A., Verschelde, J.: PHCmaple: A Maple interface to the numerical homotopy algorithms in PHCpack. In: Tran, Q.-N. (ed.) Proceedings of the Tenth International Conference on Applications of Computer Algebra (ACA 2004), pp. 139–147 (2004)
Leykin, A., Verschelde, J.: Factoring solution sets of polynomial systems in parallel. In: Skeie, T., Yang, C.-S. (eds.) Proceedings of the 2005 International Conference on Parallel Processing Workshops, Oslo, Norway, June 14-17, 2005. High Performance Scientific and Engineering Computing, pp. 173–180. IEEE Computer Society Press, Los Alamitos (2005)
Leykin, A., Verschelde, J., Zhuang, Y.: Parallel homotopy algorithms to solve polynomial systems. In: Iglesias, A., Takayama, N. (eds.) ICMS 2006. LNCS, vol. 4151, pp. 225–234. Springer, Heidelberg (2006)
Leykin, A., Verschelde, J., Zhao, A.: Newton’s method with deflation for isolated singularities of polynomial systems. Theoretical Computer Science (to appear)
Leykin, A., Verschelde, J., Zhao, A.: Evaluation of Jacobian matrices for Newton’s method with deflation to approximate isolated singular solutions of polynomial systems. In: Wang, D., Zhi, L. (eds.) SNC 2005 Proceedings. International Workshop on Symbolic-Numeric Computation, Xi’an, China, July 19-21, 2005, pp. 19–28 (2005)
Li, T.Y.: Numerical solution of polynomial systems by homotopy continuation methods. In: Cucker, F. (ed.) Handbook of Numerical Analysis. Foundations of Computational Mathematics, vol. XI, pp. 209–304. North-Holland, Amsterdam (2003)
Maekawa, M., Noro, M., Ohara, K., Okutani, Y., Takayama, N.: Openxm – an open system to integrate mathematical softwares. available at: http://www.OpenXM.org/
Maekawa, M., Noro, M., Ohara, K., Takayama, N., Tamura, Y.: The design and implementation of OpenXM-RFC 100 and 101. In: Shirayanagi, K., Yokoyama, K. (eds.) Computer mathematics. Proceedings of the Fifth Asian Symposium (ASCM 2001) Matsuyama, 26 - 28 September. Lecture Notes Series on Computing, vol. 9, pp. 102–111. World Scientific, Singapore (2001), available at: http://www.math.kobe-u.ac.jp/OpenXM/ascm2001/ascm2001/ascm2001.html
Maza, M.M., Reid, G.J., Scott, R., Wu, W.: On approximate triangular decomposition I. Dimension zero. In: Wang, D., Zhi, L. (eds.) SNC 2005 Proceedings. International Workshop on Symbolic-Numeric Computation, Xi’an, China, July 19-21, 2005, pp. 19–28 (2005)
Morgan, A.: Solving polynomial systems using continuation for engineering and scientific problems. Prentice-Hall, Englewood Cliffs (1987)
Noro, M.: A computer algebra system: Risa/Asir. In: Joswig, M., Takayama, N. (eds.) Algebra, Geometry, and Software Systems, pp. 147–162. Springer, Heidelberg (2003)
Reid, G., Verschelde, J., Wittkopf, A., Wu, W.: Symbolic-numeric completion of differential systems by homotopy continuation. In: Kauers, M. (ed.) Proceedings of the 2005 International Symposium on Symbolic and Algebraic Computation (ISSAC 2005), Beijing, China, July 24-27, pp. 269–276. ACM Press, New York (2005)
Snir, M., Otto, S., Huss-Lederman, S., Walker, D., Dongarra, J.: MPI - The Complete Reference, 2nd edn. The MPI Core, vol. 1. Massachusetts Institute of Technology (1998), available via: http://www-unix.mcs.anl.gov/mpi/
Sommese, A.J., Verschelde, J., Wampler, C.W.: Solving polynomial systems equation by equation (submitted for publication)
Sommese, A.J., Verschelde, J., Wampler, C.W.: Numerical decomposition of the solution sets of polynomial systems into irreducible components. SIAM J. Numer. Anal. 38(6), 2022–2046 (2001)
Sommese, A.J., Verschelde, J., Wampler, C.W.: Numerical irreducible decomposition using PHCpack. In: Joswig, M., Takayama, N. (eds.) Algebra, Geometry, and Software Systems, pp. 109–130. Springer, Heidelberg (2003)
Sommese, A.J., Verschelde, J., Wampler, C.W.: Introduction to numerical algebraic geometry. In: Solving Polynomial Equations. Foundations, Algorithms and Applications. Algorithms and Computation in Mathematics, vol. 14, pp. 301–337. Springer, Heidelberg (2005)
Sommese, A.J., Wampler, C.W.: The Numerical solution of systems of polynomials arising in engineering and science. World Scientific Press, Singapore (2005)
Stetter, H.J.: Numerical Polynomial Algebra. SIAM, Philadelphia (2004)
Sturmfels, B.: Solving Systems of Polynomial Equations. In: AMS 2002. CBMS Regional Conference Series in Mathematics, vol. 97 (2002)
Verschelde, J.: Algorithm 795: PHCpack: A general-purpose solver for polynomial systems by homotopy continuation. ACM Trans. Math. Softw. 25(2), 251–276 (1999), Software available at: http://www.math.uic.edu/~jan
Verschelde, J., Wang, Y.: Computing dynamic output feedback laws. IEEE Transactions on Automatic Control 49(8), 1393–1397 (2004)
Verschelde, J., Wang, Y.: Computing feedback laws for linear systems with a parallel Pieri homotopy. In: Yang, Y. (ed.) Proceedings of the 2004 International Conference on Parallel Processing Workshops, Montreal, Quebec, Canada, August 15-18, 2004. High Performance Scientific and Engineering Computing, pp. 222–229. IEEE Computer Society, Los Alamitos (2004)
Verschelde, J., Zhuang, Y.: Parallel implementation of the polyhedral homotopy method. In: The proceedings of The 8th Workshop on High Performance Scientific and Engineering Computing (HPSEC 2006), Columbus, Ohio, USA, August 18, 2006 (accepted for publication)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Leykin, A., Verschelde, J. (2006). Interfacing with the Numerical Homotopy Algorithms in PHCpack . In: Iglesias, A., Takayama, N. (eds) Mathematical Software - ICMS 2006. ICMS 2006. Lecture Notes in Computer Science, vol 4151. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11832225_35
Download citation
DOI: https://doi.org/10.1007/11832225_35
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-38084-9
Online ISBN: 978-3-540-38086-3
eBook Packages: Computer ScienceComputer Science (R0)